Kevin had five rectangular photographs. The ten side lengths of the photographs were 1 cm, 2 cm, 3 cm, …, 10 cm, (with each occurring once).
Kevin assembled the five photographs with no gaps or overlaps into a square frame of side length 11 cm. Three of the photographs had size 1 cm x 6 cm, 2 cm x 10 cm and 3 cm x 9cm.
a. Find the side lengths of the other two photographs and show how the photographs can be put in the frame.
b. Katie also had five rectangular photographs with side lengths 1 cm, 2 cm, 3 cm, … , 10 cm (with each occurring once ). She assembled hers into a rectangular frame of size 15 cm x 10 cm.
Find the side lengths of each photograph and show how the photographs can be put in the frame.
Kevin and Katie noticed that, whenever they put the photographs into a rectangular frame, they were always arranged with one photograph not touching any side of the frame.
A frame of this kind is shown.
The letters a, b, c, d, f, g, h, I, j and k represent the lengths of the photographs in centimetres and these are the numbers 1, 2, 3, …., 10 in some order.
c. Show that one of j and k is even and the other odd.
d. Show that at least one of the lengths or the width of the frame must be odd.